The Research And Construction Of Orthomorphic Permutation

Orthomorphic permutations have good characteristics and very important applications in cryptosystems, especially in the design of block cipher. In this paper, we present a notion of orthomorphic translation, based on the study of the consructure and properties of orthomorphic permutations, and draw a conclusion that the orthomorphic translations of an orthomorphic permutation are still orthomorphic permutations. Furthermore, we point out the relation of them and the equivalence class of the orthomorphic permutation. We also discuss the one-to-one correspondence of orthomorphic permutations and orthomorphic permutation polynomials, and show that for any integer m=2, ax3+bx2+cx+d(a?0) is not an orthomorphic permutation polynomial of GF(2m), according to the theory of permutation polynomials.It is a hotspot in the study of cryptography now how to construct orthomorphic permutations. In this paper, we introduce the present methods of constructing orthomorphic permutations. Moreover, we illustrate some new methods and give many examples. Firstly, we point out that the orthomorphic translations of a nonlinear orthomorphic permutation can be new nonlinear ones. Secondly, we show that how to change one orthomorphic permutation partly into another orthomorphic permutation. Thirdly, we explain that how to get an orthomorphic permutation with 2n+1-degree from one with 2n-degree, consequently leading to the result that for n > m > 1, the number of orthomorphic permutations with 2n-degree is greater than that of those with 2n-degree. Finally, we prove that by combining some orthomorphic permutations with low degree, one with an arbitrarily high degree can be obtained.About the applications of orthomorphic permutations, we introduce the method of strengthening block cipher and constructing boolean functions with perfect cryptographic characteristics, and give a scheme of enhancing the security of a cipher by the use of orthomorphic permutations.It is hard to search all the orthomorphic permutations with a high degree. In this paper, we find a quick algorithm to search them, based on the perfectly balanced property of orthomorphic permutations. By using the quick algorithm we get all the orthomorphic permutations with 16-degree, and analyse the statistic properties such as the number of linear structure, differential uniform. According to the statistic results, there is no almost perfectly nonliear (i.e. APN) orthomorphic permutation of 16-degree. Furthermore, we present a suppose that there is no APN orthomorphic permutation.